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This paper investigates robust versions of the general empirical riskminimization algorithm, one of the core techniques underlying modernstatistical methods. Success of the empirical risk minimization is based on thefact that for a "well-behaved" stochastic process $\left\{ f(X), \ f\in\mathcal F\right\}$ indexed by a class of functions $f\in \mathcal F$, averages$\frac{1}{N}\sum_{j=1}^N f(X_j)$ evaluated over a sample $X_1,\ldots,X_N$ ofi.i.d. copies of $X$ provide good approximation to the expectations $\mathbb Ef(X)$ uniformly over large classes $f\in \mathcal F$. However, this might nolonger be true if the marginal distributions of the process are heavy-tailed orif the sample contains outliers. We propose a version of empirical riskminimization based on the idea of replacing sample averages by robust proxiesof the expectation, and obtain high-confidence bounds for the excess risk ofresulting estimators. In particular, we show that the excess risk of robustestimators can converge to $0$ at fast rates with respect to the sample size.We discuss implications of the main results to the linear and logisticregression problems, and evaluate the numerical performance of proposed methodson simulated and real data.

Excess risk bounds in robust empirical risk minimization